511 research outputs found
New Solutions to the Firing Squad Synchronization Problems for Neural and Hyperdag P Systems
We propose two uniform solutions to an open question: the Firing Squad
Synchronization Problem (FSSP), for hyperdag and symmetric neural P systems,
with anonymous cells. Our solutions take e_c+5 and 6e_c+7 steps, respectively,
where e_c is the eccentricity of the commander cell of the dag or digraph
underlying these P systems. The first and fast solution is based on a novel
proposal, which dynamically extends P systems with mobile channels. The second
solution is substantially longer, but is solely based on classical rules and
static channels. In contrast to the previous solutions, which work for
tree-based P systems, our solutions synchronize to any subset of the underlying
digraph; and do not require membrane polarizations or conditional rules, but
require states, as typically used in hyperdag and neural P systems
The Firing Squad Synchronization Problems for Number Patterns on a Seven-Segment Display and Segment Arrays
The Firing Squad Synchronization Problem (FSSP), one of the most well-known problems related to cellular automata, was originally proposed by Myhill in 1957 and became famous through the work of Moore [1]. The first solution to this problem was given by Minsky and McCarthy [2] and a minimal time solution was given by Goto [3]. A significant amount of research has also dealt with variants of this problem. In this paper, from a theoretical interest, we will extend this problem to number patterns on a seven-segment display. Some of these problems can be generalized as the FSSP for some special trees called segment trees. The FSSP for segment trees can be reduced to a FSSP for a one-dimensional array divided evenly by joint cells that we call segment array. We will give algorithms to solve the FSSPs for this segment array and other number patterns, respectively. Moreover, we will clarify the minimal time to solve these problems and show that there exists no such solution
Exploring Millions of 6-State FSSP Solutions: the Formal Notion of Local CA Simulation
In this paper, we come back on the notion of local simulation allowing to
transform a cellular automaton into a closely related one with different local
encoding of information. This notion is used to explore solutions of the Firing
Squad Synchronization Problem that are minimal both in time (2n -- 2 for n
cells) and, up to current knowledge, also in states (6 states). While only one
such solution was proposed by Mazoyer since 1987, 718 new solutions have been
generated by Clergue, Verel and Formenti in 2018 with a cluster of machines. We
show here that, starting from existing solutions, it is possible to generate
millions of such solutions using local simulations using a single common
personal computer
A Minimal Time Solution to the Firing Squad Synchronization Problem with Von Neumann Neighborhood of Extent 2
Cellular automata provide a simple environment in which to study global behaviors. One example of a problem that utilizes cellular automata is the Firing Squad Synchronization Problem, first proposed in 1957. This paper provides an overview of the standard Firing Squad Synchronization Problem and a commonly used technique in solving it. This paper also provides a statement of a new extension of the Standard Firing Squad Synchronization Problem to a different neighborhood definition - a Von Neumann neighborhood of extent 2. An 8 state 651 rule minimal time solution to the extended problem is described, presented and proven, along with Python code used in running simulations of the solution
A Simple Optimum-Time FSSP Algorithm for Multi-Dimensional Cellular Automata
The firing squad synchronization problem (FSSP) on cellular automata has been
studied extensively for more than forty years, and a rich variety of
synchronization algorithms have been proposed for not only one-dimensional
arrays but two-dimensional arrays. In the present paper, we propose a simple
recursive-halving based optimum-time synchronization algorithm that can
synchronize any rectangle arrays of size m*n with a general at one corner in
m+n+max(m, n)-3 steps. The algorithm is a natural expansion of the well-known
FSSP algorithm proposed by Balzer [1967], Gerken [1987], and Waksman [1966] and
it can be easily expanded to three-dimensional arrays, even to
multi-dimensional arrays with a general at any position of the array.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
A Genetically Evolved Solution to the Firing Squad Problem
In 1957, J. Myhill presented the firing squad problem. A special case of k-color cellular automata (CA) synchronization, the firing squad problem offers more stringent rules allowing for a provable minimal running time. To date, CA solutions have been found that run in minimal time using as many as sixteen states and as few as six [5]. There have also been arguments against the existence of solutions using only 4 states [11]. Due to the extremely large search space involved with such problems, the existing solutions have all been analytic in nature. We attempt to apply genetic algorithms and genetic programming to create transition tables that solve the firing squad problem. Ideally, the solutions would run in minimal time. No generalized solutions were found, but progress was made towards determining the best strategies for an evolved solution
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